Entanglement detection in hybrid optomechanical systems

We study a device formed by a Bose-Einstein condensate (BEC) coupled to the field of a cavity with a moving end mirror and find a working point such that the mirror-light entanglement is reproduced by the BEC-light quantum correlations. This provides an experimentally viable tool for inferring mirror-light entanglement with only a limited set of assumptions. We prove the existence of tripartite entanglement in the hybrid device, persisting up to temperatures of a few milli-Kelvin, and discuss a scheme to detect it.

Figure 1

Sketch of the detection scheme. The probe field has the same periodicity of the cavity field. It impinges on a semireflecting mirror, creating a standing wave. Part of the probe light is transmitted and then measured with the output light of the cavity.

Figure 2

(a) Logarithmic negativity $E_{\mathit{AC}}$ against $\Delta$ (in units of ${\omega }_m/2\pi =3\times {10}^6$ ${\mathrm{s}}^{-1}$) and the cavity-mirror coupling rate $\chi$. (b) Same as in (a), but for $E_{\mathit{MC}}$. We have taken $T=10$ $\mu$K, mechanical quality factor $3\times {10}^4$, $m=50$ ng, $\mathcal{R}=50$ mW, cavity finesse $F={10}^4$, and $\zeta =\chi$. The cavity decay rate is given by $\kappa =\pi c/2LF$, where $c$ is the speed of light and the cavity length is $L=1$ mm. (c) Comparison between $E_{\mathit{AC}}$ (solid line) and $E_{\mathit{MC}}$ (dotted line) against $T$ for $\Delta =2{\omega }_m$ and $\chi =100$ ${\mathrm{s}}^{-1}$. Other parameters are as in (a).

Figure 3

(a) Logarithmic negativity $E_{\mathit{AC}}$ (solid line) and $E_{\mathit{MC}}$ (dashed line) against $\zeta /\chi$ for $\Delta =2{\omega }_m$. Other parameters are as in Fig. 2. (b) $E_{\mathit{AC}}$ (solid line) and $E_{\mathit{MC}}$ (dashed line) against $\Omega /{\omega }_m$. The inset shows the entanglement functions around $\Omega ={\omega }_m$ for $T=1$ $\mu$K and $F=4\times {10}^4$.

Figure 4

$E_{i|\mathit{jk}}$ ($i,j,k=A,M,C$) and the genuine tripartite entanglement $G_{\mathrm{tri}}$ against $T$ for $\Omega ={\omega }_m$ and $\chi =\zeta$. The lines are $E_{A|\mathit{MC}}=E_{M|\mathit{AC}}$, solid line; $E_{C|\mathit{AM}}$, dashed line; and $G_{\mathrm{tri}}$, dot-dashed line. Other parameters are as in Fig. 2.